The introduction starts with a story behind the conception of J.S. Bach’s Musical Offering. Bach went for an impromptu visit to Frederick the Great, King of Prussia, and was given the chance to try out the King’s collection of pianos while improvising on a theme composed by his Highness. After the visit, Bach undertook the creation of Musical Offering which eventually became one of his most notable works. The book’s author, Douglas Hofstadter, then discussed about the musical structures that Bach’s Musical Offering contained. He defined canons as those having similar sets of notes played in harmony with each other by the use of varying tones, tempos, time of attacks or even by inversion of notes. As for fugues, they are just like canons but having more freedom for musical improvisation.

What caught DH’s attention was that canons and fugues have similar properties with the central concept of the book, that of strange loops. A strange loop is a hierarchy of complexities such that as one moves up or down the levels, it will find itself back to the same level of complexity it initially assumed. Now a canon or a fugue behaves like strange loops because of the recurring set of notes throughout the musical piece. The two are differentiated such that a canon strictly obeys the structure of a strange loop, a fugue has more freedom to play around other notes thus instilling it with creativity within its own strange loop. Another example that illustrates strange loops is that of Epimenides’ paradox: The following statement is false. The preceding sentence is true. This is a strange loop that goes back to its initial level of hierarchy in a series of two steps. Other complex systems that exhibit strange loopiness will be discussed further in the book.

Strange loops can also be understood with the help of the mathematically-inspired artist Maurits Cornelis Escher. Below is his art entitled Drawing Hands where it has the same two-level hierarchy as that of Epimenides’ Paradox discussed above.

Another of the strange loopiness is Escher’s Waterfall depicted below. It undergoes six levels of hierarchy before looping back on itself.

From here we arrive at the third and probably the most important in the three personas that make up GEB, the mathematician/logician Kurt Godel and his celebrated incompleteness theorem. To aid the reader in understanding the theorem further, the book took a bit of a recourse to expound on the history of mathematical logic and the motives behind Russell and Whitehead’s logical magnum opus, the Principia Mathematica. The goal was to make logic, set theory and number theory free from self-reference; free from the perils of strange loops. To be able to prove any statement within a system by use of other statements belonging to the same system was the ultimate quest for mathematical consistency and completeness. This was to put mathematics on a firm foundation, its inductive power envisioned to be free from senselessness. But not for long. It was Kurt Godel who put a stump on Russell and Whitehead and all the other mathematicians who put their hopes in the Principia Mathematica. Godel’s merit was due to his incompleteness theorem which, in layman’s terms, is as follows:

“All consistent axiomatic formulations of number theory include undecidable propositions.”

Why number theory? Because prior to the incompleteness theorem, Godel first showed that any symbol, statement, or formula in some formal language can be assigned a unique natural number. This process is called Godel numbering. It is through Godel’s numbering that, in a sense, all of mathematics can be reduced to the study of number theory. That’s why it was number theory that Godel has attacked forthrightly in his incompleteness theorem. This shows that one still cannot prove all the statements of a given system even if the system had perfectly implemented the rules written in the Principia Mathematica. There is no escape to self-reference, any formal system will always arrive at inconsistencies and become incomplete. No axiomatic system whatsoever could produce all number-theoretical truths, unless it were an inconsisten system! Moreover, Godel showed that provability is a weaker notion than truth. This means that self-referential statements are not necessarily false, they are just unprovable. (There is also an analogue of Godel’s theorem in the field of computing, formulated by Alan Turing).

After the introduction of the three personas (Godel, Escher, Bach), the book proceeds to link these three together in the spirit of strange loops. DH has called this synthesis as an “Eternal Golden Braid”. It is here that he introduces the first dialogue of the book entitled Three-Part Invention. The aim of the dialogues is to stir up the reader’s familiarity with self-referring frameworks. It intends to revert the reader, in encountering strange loops, from intuitively saying “This doesn’t make sense. This is wrong.” to a humbler and logically honest stance.

References:
Hofstadter, Douglas R. (1999) [1979], Gödel, Escher, Bach: An Eternal Golden Braid, Basic Books.

http://en.wikipedia.org/wiki/File:DrawingHands.jpg

http://en.wikipedia.org/wiki/File:Escher_Waterfall.jpg

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